Question

If \(p<x<q\) and \(r<y<s\), is \(x>y\)?
I. p = r
II. q<r

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In inequality problems, try to combine the given information into a single chain. If you can establish a direct relationship like \(x<y\) or \(x>y\), the information is sufficient. If the ranges overlap, the information is usually not sufficient.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a Data Sufficiency question involving inequalities. We need to determine if we can definitively answer "yes" or "no" to the question "is \(x>y\)?". The initial information establishes ranges for \(x\) and \(y\).
Step 2: Detailed Explanation:
Analyze Statement I: "p = r"
This tells us that the lower bounds of the ranges for \(x\) and \(y\) are the same. Let's take an example: Let \(p=r=5\), \(q=10\), and \(s=12\). So we have \(5<x<10\) and \(5<y<12\).

Can \(x>y\)? Yes. For example, if \(x=7\) and \(y=6\).
Can \(x \leq y\)? Yes. For example, if \(x=7\) and \(y=8\).
Since we can get both "yes" and "no" answers, Statement I alone is not sufficient.
Analyze Statement II: "q<r"
This statement tells us that the upper bound for \(x\) (\(q\)) is less than the lower bound for \(y\) (\(r\)).
We are given: \[ x<q \quad \text{and} \quad r<y \] Combining these with the information from the statement (\(q<r\)), we get a chain of inequalities: \[ x<q<r<y \] From this chain, we can definitively conclude that \(x<y\).
This provides a definitive "no" to the question "is \(x>y\)?". Therefore, Statement II alone is sufficient.
Step 3: Final Answer:
Statement II alone is sufficient to answer the question, but statement I alone is not. This corresponds to option (B).